A260990 -id:A260990 - cp's OEIS frontend

A260991 Numbers n such that the modular curve X_0(n) has a bielliptic involution of Atkin-Lehner type.

22, 26, 28, 30, 33, 34, 35, 37, 38, 39, 40, 42, 43, 44, 45, 48, 50, 51, 53, 54, 55, 56, 60, 61, 62, 63, 64, 65, 69, 75, 79, 81, 83, 89, 92, 94, 95, 101, 119, 131

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Same as A260990 except that now 72 is excluded.

Cf. A260992, A260993, A260994.

A260992 Numbers n such that the modular curves X(n) and X_1(n) are not bielliptic.

Original entry on oeis.org

52, 57, 58, 66, 67, 68, 70, 73, 74, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 93, 96, 97, 98, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 132, 133, 134

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

nonn

The list includes all n >= 132.

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Cf. A260990, A260991.

A260993 Numbers n such that the modular curve X_0(n) contains infinitely many rational points of degree 2.

Original entry on oeis.org

22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 43, 46, 47, 48, 50, 53, 59, 61, 65, 71, 79, 83, 89, 101, 131

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Cf. A260990, A260991, A260992, A260994.

A260994 Numbers n such that the modular curve X_0(n) has genus >= 2 and contains infinitely many points of degree 2 over some number field L.

Original entry on oeis.org

22, 23, 26, 28, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 53, 54, 55, 56, 59, 60, 61, 62, 63, 64, 65, 69, 71, 72, 75, 79, 81, 83, 89, 92, 94, 95, 101, 119, 131

Offset: 1

N. J. A. Sloane, Aug 08 2015, following a suggestion from Roger L. Bagula, Jul 22 2015

Francesc Bars, Bielliptic Modular Curves, Journal of Number Theory, 76 (1999), 154-165.

Cf. A260990, A260991, A260992, A260993.

A276182 Numbers N such that the modular curve X_0(N) is hyperelliptic.

Original entry on oeis.org

22, 23, 26, 28, 29, 30, 31, 33, 35, 37, 39, 40, 41, 46, 47, 48, 50, 59, 71

Offset: 1

Gheorghe Coserea, Oct 17 2016

"The only case where the hyperelliptic involution is not defined by an element of SL(2, R) is N=37."
"For N = 40, 48 the hyperelliptic involution v is not of Atkin-Lehner type. The remaining sixteen values are listed in the table below, together with their genera and hyperelliptic involutions v." (see Ogg link)
n    N    g    v
  22   2    11
  23   2    23
  26   2    26
  28   2    7
  29   2    29
  30   3    15
  31   2    31
  33   3    11
  35   3    35
 39   3    39
 41   3    41
 46   5    23
 47   4    47
 50   2    50
 59   5    59
 71   6    71

J. S. Balakrishnan, B. Mazur, and N. Dogra, Ogg's torsion conjecture: fifty years later, Bull. Amer. Math. Soc., 62:2 (2025), 235-268.

Andrew P. Ogg, Hyperelliptic modular curves, Bulletin de la S. M. F., 102 (1974), p. 449-462.

Cf. A001617, A260990.

cp's OEIS Frontend

A260991 Numbers n such that the modular curve X_0(n) has a bielliptic involution of Atkin-Lehner type.

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A260992 Numbers n such that the modular curves X(n) and X_1(n) are not bielliptic.

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A260993 Numbers n such that the modular curve X_0(n) contains infinitely many rational points of degree 2.

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A260994 Numbers n such that the modular curve X_0(n) has genus >= 2 and contains infinitely many points of degree 2 over some number field L.

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A276182 Numbers N such that the modular curve X_0(N) is hyperelliptic.

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